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Quick Updates for the Perturbed Static Output Feedback Control Problem in Linear Systems with Applications to Power Systems

MirSaleh Bahavarnia, Ahmad F. Taha

TL;DR

This work tackles the challenge of updating a nominal stabilizing static output feedback (SOF) controller when a perturbed linear model arises, common in power systems. It replaces costly re-synthesis with a minimum destabilizing real perturbation (MDRP)-based norm-minimization, yielding a fast update F^{updated} = F^{nominal} + G^* where G^* minimizes ||BGC + Δ||_F. The authors derive bounds and a parameterization that lead to guaranteed stability regions and a geometric distance-to-instability metric, and they validate the approach through toy and power-system case studies showing orders-of-magnitude speedups with comparable stabilization performance to SDP-based methods. The results demonstrate practical applicability for frequent operating-point updates in large-scale networks, while also acknowledging NP-hardness of exact MDRP and relying on effective heuristics for real-time updates.

Abstract

This paper introduces a method for efficiently updating a nominal stabilizing static output feedback (SOF) controller in perturbed linear systems. As operating points and state-space matrices change in dynamic systems, accommodating updates to the SOF controller are necessary. Traditional methods address such changes by re-solving for the updated SOF gain, which is often (i) computationally expensive due to the NP-hard nature of the problem or (ii) infeasible due to the limitations of its semidefinite programming relaxations. To overcome this, we leverage the concept of minimum destabilizing real perturbation (MDRP) to formulate a norm minimization problem that yields fast, reliable controller updates. This approach accommodates a variety of known perturbations, including abrupt changes, model inaccuracies, and equilibrium-dependent linearizations. We remark that the application of our proposed approach is limited to the class of SOF controllers in perturbed linear systems. We also introduce geometric metrics to quantify the proximity to instability and rigorously define stability-guaranteed regions. Extensive numerical simulations validate the efficiency and robustness of the proposed method. Moreover, such extensive numerical simulations corroborate that although we utilize a heuristic optimization method to compute the MDRP, it performs quite well in practice compared to an existing approximation method in the literature, namely the hybrid expansion-contraction (HEC) method. We demonstrate the results on the SOF control of multi-machine power networks with changing operating points, and demonstrate that the computed quick updates produce comparable solutions to the traditional SOF ones, while requiring orders of magnitude less computational time.

Quick Updates for the Perturbed Static Output Feedback Control Problem in Linear Systems with Applications to Power Systems

TL;DR

This work tackles the challenge of updating a nominal stabilizing static output feedback (SOF) controller when a perturbed linear model arises, common in power systems. It replaces costly re-synthesis with a minimum destabilizing real perturbation (MDRP)-based norm-minimization, yielding a fast update F^{updated} = F^{nominal} + G^* where G^* minimizes ||BGC + Δ||_F. The authors derive bounds and a parameterization that lead to guaranteed stability regions and a geometric distance-to-instability metric, and they validate the approach through toy and power-system case studies showing orders-of-magnitude speedups with comparable stabilization performance to SDP-based methods. The results demonstrate practical applicability for frequent operating-point updates in large-scale networks, while also acknowledging NP-hardness of exact MDRP and relying on effective heuristics for real-time updates.

Abstract

This paper introduces a method for efficiently updating a nominal stabilizing static output feedback (SOF) controller in perturbed linear systems. As operating points and state-space matrices change in dynamic systems, accommodating updates to the SOF controller are necessary. Traditional methods address such changes by re-solving for the updated SOF gain, which is often (i) computationally expensive due to the NP-hard nature of the problem or (ii) infeasible due to the limitations of its semidefinite programming relaxations. To overcome this, we leverage the concept of minimum destabilizing real perturbation (MDRP) to formulate a norm minimization problem that yields fast, reliable controller updates. This approach accommodates a variety of known perturbations, including abrupt changes, model inaccuracies, and equilibrium-dependent linearizations. We remark that the application of our proposed approach is limited to the class of SOF controllers in perturbed linear systems. We also introduce geometric metrics to quantify the proximity to instability and rigorously define stability-guaranteed regions. Extensive numerical simulations validate the efficiency and robustness of the proposed method. Moreover, such extensive numerical simulations corroborate that although we utilize a heuristic optimization method to compute the MDRP, it performs quite well in practice compared to an existing approximation method in the literature, namely the hybrid expansion-contraction (HEC) method. We demonstrate the results on the SOF control of multi-machine power networks with changing operating points, and demonstrate that the computed quick updates produce comparable solutions to the traditional SOF ones, while requiring orders of magnitude less computational time.
Paper Structure (21 sections, 5 theorems, 70 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 5 theorems, 70 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction and Paper Contributions
  2. Problem Statement
  3. Updating the Nominal Stabilizing SOF Controller
  4. Motivation
  5. Lower bound
  6. Upper bound
  7. Exact value
  8. Main idea of the updated gain
  9. Main Results
  10. Norm-bounded perturbation analytic parameterization
  11. Guaranteed stability region analytic characterization
  12. Numerical Simulations
  13. On the computation of the MDRP
  14. A toy example
  15. Two power system examples
  16. ...and 6 more sections

Key Result

Lemma 1

Suppose that $H = U_H \Sigma_H V_H^\top$ is the SVD of $H$. Then, $P$ in ROV can be parameterized as follows: where $U_H = (V_C \otimes U_B)U_{\Omega}$ holds provided that $B = U_B \Sigma_B V_B^\top$, $C = U_C \Sigma_C V_C^\top$, and $\Omega := \Sigma_C^\top \otimes \Sigma_B = U_{\Omega} \Sigma_{\Omega} V_{\Omega}^\top$ denote the SVDs of $B$, $C$, and $\Omega$, respectively.

Figures (6)

  • Figure 1: Dependency of $\frac{J^{\ast}(\Delta)}{\rho^2}$ on $\tau$ and $\theta$.
  • Figure 2: (a) The guaranteed stability region $S_{\kappa}$ for $\kappa = \frac{1}{3}$, (b) the percentage-based lower bound on the stability of the updated perturbed state-space \ref{['RDPerturbedSD']}$\xi_{\kappa}~(\%)$ versus $\kappa$, (c) the percentage-based lower bound on the stability of the updated perturbed state-space \ref{['RDPerturbedSD']}$\xi_{\rho}~(\%)$ versus $\rho$ for $\beta = 1$, and (d) the percentage-based lower bound on the stability of the updated perturbed state-space \ref{['RDPerturbedSD']}$\xi_{\beta}~(\%)$ versus $\beta$ for $\rho = 1$.
  • Figure 3: Stability regions for $AC3$ benchmark with (a) $\rho = 2\beta^{\mathrm{accurate}}$ and $\beta = \beta^{\mathrm{accurate}}$ (computed via solving the optimization problem \ref{['OPaux']}) and (b) $\rho = 2\beta^{\mathrm{accurate}}$ and $\beta = \beta^{\mathrm{inaccurate}}$: the guaranteed (conservative) stability regions based on Proposition \ref{['Prop2']} (filled with blue circles) and the exact ones based on $\alpha(A+ BF^{\text{nominal}}C + \Delta + B G_{\Delta}^{\ast}C) = \alpha(A+\Delta + B F^{\text{updated}} C) < 0$ with $G_{\Delta}^{\ast}$ in \ref{['Osol']} (filled with red asterisks).
  • Figure 4: Relative error of control inputs $100 \times \frac{\|u^{\text{updated}}(t)-u^{\text{typical}}(t)\|}{\|u^{\text{typical}}(t)\|}$ ---with $u^{\text{updated}}(t) = F^{\text{updated}} y^{\text{updated}}(t)$ and $u^{\text{typical}}(t) = F^{\text{typical}} y^{\text{typical}}(t)$ as control inputs considering case $14$-bus.
  • Figure 5: Spectra associated with $A+\Delta$, $A+\Delta + BF^{\text{updated}}C$, and $A+\Delta + BF^{\text{typical}}C$ considering the three IEEE power system test cases: (a) case $9$-bus, (b) case $14$-bus, and (c) case $39$-bus.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Corollary 1
  • Proposition 2
  • proof
  • Corollary 2